Geometric Sequences / First Day
Hey guys, it's Manny doing the scribe for today, cause yeah Michael couldn't so I volounteered for it. Today, we started our new unit Geometric Sequences, and thankfully, or sadly for some, it's our last. It was a lot to absorb without taking some form of notes, so I'll try my best to describe it to you guys.
So what is a sequence? A sequence is a list of numbers that follows a certain rule. That's what it is.
In the beginning of class, Mr.K put up a question with 6 parts that looked like this:
Find the next Four Terms in Each Sequence:
a) 4, 7, 10, 13, __, __, __, __
b) 3, 6, 12, 24, __, __, __, __
c) 1, 2, 4,__, __, __, __
d) 2, 5, 10, 17, 26, __, __, __, __
e) 2, 1, 3, 4, 7, 11 __, __, __, __
f) 77, 49, 36, 18, __, __, __, __
(We talked a lot about the first two.)
Well, at first we thought it was real easy, something done in like grade five. The first one (part a) we said, add 3 to the last number, easy as cutting butter. Well it was, but why is it 3? We all went speechless. Then we said that because taking two that are beside eachother and finding the difference between them will give us 3, so therefore it always has to go up by 3. So this is what we did.
7 - 4 = 3
10 - 7 = 3
13 - 10 = 3
And it just so happens that the 3 is called the common difference.
So this is what we have so far for part a) 4, 7, 10, 13, 16, 19, 22, 25.
Then, Mr. K, asked so what would the 11th number be? We answered 34. It was right, and we did this by adding 3, to the last number three more times, since there are eight terms already.
It turns out Mr.K made a "mistake," he meant to ask for the 1001th number. Well the 11th, and 1001th, or n in this case, are called ranks. One of us saw that it was one number less than the rank, multiplied by 3 and added to the first terms which was 4. The term is the output of the rank.
We came up with the formula tn = t1 + d(n-1).
Where tn is the term of the number (or rank number),and Where t1 is the first term, and
Where d is the common difference.
So therefore, tn = 4 + 3(n-1).
And for the problem for the 1001th looks like this t1001 = 4 + 3(1001-1), which equals out to a huge number bigger than googal =] (There is not as much atoms in the universe than the number googal 10100).
This type of equation has a name called the Recursive Definition, because recursive means to apply some feature over and over again, here we are adding 3 all the time.
Well now that we know all this information, what is the initial term, Mr. K asks? Well since we added 3 to the last number to get the next, we can just subtract 3 from the first. And at the 0th rank, we have 1. This is what it looks like.
Since we now know the 0th term, we can now write another equation. And come up with tn = dn + to, therefore being tn = 3n + 1. This equation is called the Impicit Definition, because implicit means that is implied, not obvious, and is in the subtext kind of like when some of us nodd or a raise an eyebrow to say hello to a friend while an adult would not understand it.
Then we graphed it to look like this.
We did this to show that from this pattern, given only 4 pieces of info, we can show it
symbollically - written as an equation
numerically - table of values
graphically -drawn out as a graph
Which is shown above. These are the 3 perspectives you can look at just about anything given at least one of them.
We are only learning about two types of sequences in this course. Arithmetic, and geometric. Arithmetic deals in adding/subtracting types of patterns, which was what we just did. Geometric is dealing with multiplication/division types of patterns.
Part b) 3, 6, 12, 24, __, __, __, __, is geometric.
- Will explain soon -
If anyone is curious the answers to the question above are:
a) 4, 7, 10, 13, 16, 19, 22, 25 [common difference 3]
b) 3, 6, 12, 24, 48, 96, 192, 384 [common ratio 2]
c) 1, 2, 4, 8, 16, 32, 64 [powers of 2 raised to the exponent of the (rank - 1)]
d) 2, 5, 10, 17, 26, 37, 50, 55, 72 [square the rank + 1]
e) 2, 1, 3, 4, 7, 11, 18, 29, 37 [sum of two side by side numbers, or fibonacci sequence]
f) 77, 49, 36, 18, 8, 8, 8, 8 [multiply digits]
We only did the first two today, and will not be looking at the last three parts, as they are different from arithmetic and geometric sequences.
Well, I have had a long day today. I was asked to volounteer last minute, and from the kindness of my heart I couldn't refuse, just like how I volounteered for this scribe today. But I am tired, and starving, I have had nothing but junk food since the past 14 hours, and it's way past my bedtime. I will continue the post tomorrow morning, and if not by the end of tomorrow. I'm sorry if this causes an inconvience to anybody and will be happy to fill you in if you want face to face.
The scribe for tomorrow's class will I guess be Michael since he didn't do it today, and if that's not possible, it'll be Master Teddie ;).